Acoustoelastic Theory Research Articles

Semi-analytical peridynamic method for modal analysis of acoustoelastic Lamb waves

Lamb wave has been widely used as a non-destructive testing tool for inspecting the defects or damage in the plate system. A comprehensive understanding and correct prediction of the modal characteristics of Lamb waves are of high importance for ensuring successful practical applications. In this paper, a new method called the semi-analytical peridynamic (SAPD) method for analyzing wave propagation is developed. This method, within the framework of the general acoustoelasticity theory, uses the peridynamic differential operator to transform the equations of motion for guided waves in prestressed anisotropic media and the boundary conditions from local differential forms to nonlocal integral forms. By introducing meshfree discretization and Lagrange multipliers, these governing equations can be reorganized into a standard generalized eigenvalue formalism and solved. The effectiveness and accuracy of the SAPD method are first verified through comparison with the exact solutions. Phase and group velocity dispersion curves and displacement distributions of Lamb waves in three typical cases are then calculated to study the effects of material heterogeneity, applied stress and residual stress on the propagation of Lamb waves. Since complex grid generation algorithms are avoided, the SAPD method exhibits the advantages in terms of simplicity and implementation.

Acoustoelastic Theory and Mode Analysis of Bolted Structures Under Preload

Bolted connections are a common feature of connection in mechanical structures, employed to secure connected parts by tightening nuts and providing preload. The preload is susceptible to various factors leading to potential bolt loosening. The acoustoelastic theory is the most common measure of a bolt structure’s stress. The present study investigates the relationship between the inherent properties of a structure and its acousticelastic properties. The modal response of the bolted structure under different preload forces is studied by translating the acoustoelastic relationship of the structure into an analysis of its intrinsic properties. The modal analysis reflects the relative change in wave velocity to be determined implicitly based on the eigenfrequencies of the structure. A frequency formulation of classical bolted structures based on acoustoelastic theory is presented in this paper to conduct the intrinsic characteristic analysis of bolted structures. The COMSOL5.4 simulation results are under the acoustic elasticity coefficients for ultrasonic wave propagation in bolt structures, as predicted by the acoustic elasticity theory, and the present solutions are compared with those available in the literature to confirm their validity. A systematic parameter study for bolted structures under the varying preloads with different material parameters, Lame elastic constants, Murnaghan third-order elastic constants, and structural parameters are presented. These results may serve as a benchmark for researchers in this field.

Estimation of stresses in 1D structures via inverse wavenumber analysis

The ANR ITHERATE project proposes to develop identification procedures of residual stresses in additive manufacturing via in-situ and non destructive methods using the wide band wave response of structures. This work presents a proof on concept on one-dimensional waveguides, as a preliminary results to more complex configurations. An original experimental setup is implemented using a cylindrical lens to optically amplify the vibrations of the structure in the transverse direction, allowing to resolve the wave propagation with a high-speed low-resolution camera. The dispersion of the wave velocity with the frequency, obtained via high-resolution wavenumber analysis (HRWA), is compared to predictions given by the acousto-elastic theory. In particular, the effects of the quasi-static initial stress and the elasticity of the material can be separated and estimated seperately via an inverse method that is independent from the boundary conditions and applied loads. Interestingly, this wave-based estimation does not necessitate an a-priori knowledge of the material constitutive law, overcoming the limitations of most stress estimation techniques of the state of the art, that are based on a measurement of the strain. Finally, perspectives for the generalization of the method to more complex situations, as plates under thermal loads, will be presented.

Ultrasonic wave simulation in shrink-fit assembly for the estimation of stress at the contact interference

The investigation of stress distribution in an interference fit contact region is essential information required in fatigue and wear calculations to determine design life, regrinding requirements, and maintenance schedules. The aim of this work was to use ultrasound to non-destructively determine stress in the shrink-fit assembly, with the acoustoelastic theory. This one is based on the dependence of the propagation velocity of the ultrasonic wave with the stress state in the material. When a material is subjected to stress, there is a variation of the propagation velocity of the ultrasonic wave. Three methods have been initiated to ensure that the results will be more real, the first is the analytical calculation using thick-walled cylinder theory and Lamé formulation, then a numerical modeling of the contact between the assembled parts using finite element analysis and the third one is using elastic wave simulation and acoustoelastic theory in order to determine the value of the stress distribution at the interference region.

Stress-dependent reflection and transmission of elastic waves under confining, uniaxial, and pure shear prestresses

Insights into the reflection and transmission (R/T) of waves at a prestressed interface are important in geophysical applications, such as evaluating the angle-dependent elastic properties for monitoring geopressure and tectonic stress using sonic logging data or seismic data. Although many studies deal with wave propagation in prestressed media, the angle-dependent R/T of waves at an interface subject to different prestress loading modes remains largely unaddressed. We address this issue by applying the theory of acoustoelasticity with third-order acoustoelastic constants to study the R/T coefficients at the interface between two prestressed media. Stress-induced elastic deformations are assumed to be locally homogeneous without boundary dislocations caused by stress concentration so that the static boundary conditions can be applied. We consider three typical prestress modes (confining, uniaxial, and pure shear), each of which takes into account the incidence of upgoing and downgoing P and S waves. The Knott equations under different types of prestresses are derived, followed by the estimation of angle-dependent R/T coefficients. The energy conservation at the interface and the acoustoelastic finite-difference simulation of predicted P and S modes verify the correctness of the angle-dependent R/T coefficients under confining prestress. Comparisons with the elastic case (prestress [Formula: see text]) indicate the important influence of prestresses on the energy distribution of reflected and transmitted waves, including stress-dependent critical angles, converted waves, and R/T energy ratios. Such acoustoelastic effects mainly occur around/after the critical angle. For small-angle incidence, prestresses mainly affect the gradient of R/T coefficients. The type and magnitude of prestress are closely related to the angle-dependent R/T coefficients and must be considered for amplitude-variation-with-offset analysis in prestressed media.

In Vivo Optical Coherence Elastography Unveils Spatial Variation of Human Corneal Stiffness.

The mechanical properties of corneal tissues play a crucial role in determining corneal shape and have significant implications in vision care. This study aimed to address the challenge of obtaining accurate in vivo data for the human cornea. We have developed a high-frequency optical coherence elastography (OCE) technique using shear-like antisymmetric (A0)-mode Lamb waves at frequencies above 10 kHz. By incorporating an anisotropic, nonlinear constitutive model and utilizing the acoustoelastic theory, we gained quantitative insights into the influence of corneal tension on wave speeds and elastic moduli. Our study revealed significant spatial variations in the shear modulus of the corneal stroma on healthy subjects for the first time. Over an age span from 21 to 34 (N = 6), the central corneas exhibited a mean shear modulus of 87 kPa, while the corneal periphery showed a significant decrease to 44 kPa. The central cornea's shear modulus decreases with age with a slope of -19 +/- 8 kPa per decade, whereas the periphery showed non-significant age dependence. The limbus demonstrated an increased shear modulus exceeding 100 kPa. We obtained wave displacement profiles that are consistent with highly anisotropic corneal tissues. Our approach enabled precise measurement of corneal tissue elastic moduli in situ with high precision (<7%) and high spatial resolution (<1 mm). Our results revealed significant stiffness variation from the central to peripheral corneas. The high-frequency OCE technique holds promise for biomechanical evaluation in clinical settings, providing valuable information for refractive surgeries, degenerative disorder diagnoses, and intraocular pressure assessments.

Investigating acoustoelasticity of plane elastic waves and second harmonics within isotropic solid media: A novel approach

This paper presents a novel approach to investigating the acoustoelasticity of plane elastic waves and second harmonics within isotropic solid media. The new contributions of this approach consist of providing an intuitive understanding of the essence of acoustoelasticity, offering a more straightforward method to deduce the expressions for acoustic velocities and pre-strains in natural coordinates compared to conventional theory of acoustoelasticity, and introducing a novel methodology for investigating acoustoelasticity of plane elastic waves and second harmonics. The core idea of this approach can be explained as follows: by introducing an intermediate state for describing predeformation between the natural state and the final state, the predefomation can be integrated into the nonlinear wave equation, and acoustoelastic equations of plane elastic waves and second harmonics can be deduced consequently by decomposing this nonlinear wave equation under the second approximation of perturbation theory. When applying this approach to study the acoustoelasticity of P wave and S wave of plane elastic waves, the validity and practicality of this approach are confirmed by contrasting expressions for acoustic velocities and pre-strains in natural coordinates to those found in previous research [8], which exhibit perfect consistency. Furthermore, this paper focuses on the acoustoelasticity of the free second harmonics and the driven second harmonics generated by self-excitation and interaction of plane elastic waves. The results concerning acoustoelasticity can be summarized as follows: firstly, pre-strain only influences wave vectors of plane elastic waves through the speed term in their expressions without altering their amplitudes; secondly, the acoustoelasticity of free second harmonics follows similar rules above; finally, Pre-strain affects the amplitudes and wave vectors of driven second harmonics through the speed terms in their expressions without changing the form of their expressions.

Stress amplification and relaxation imaging around cracks in nanocomposite gels using ultrasound elastography.

The quantification and modeling of gel fracture under large strain and dissipative conditions is still an open issue. In this study, a novel method for investigating the mechanical behavior of gels under highly deformed states, specifically in the vicinity of the crack tip, was developed to gain insights into fracture processes. Shear wave elastography, originally developed for the biomedical community, is employed as a powerful tool to quantitatively map the local elasticity of model gels. Here, the local stress is experimentally measured from the shear wave velocity according to nonlinear acoustoelasticity theory. The stress concentration observed at the crack tip in elastic gels is validated using classical finite element methods. Subsequently, the mechanisms of network rearrangements in viscoelastic gels (with silica nanoparticles) are analyzed both spatially and temporally. These gels consist of 90 wt% water and are synthesized with sticky nanoparticles to introduce exchangeable sacrificial bonds that facilitate stress relaxation. The nanoparticles efficiently provide stress relaxation around the crack tip, mitigating a stress singularity. The amplitude of stress relaxation was measured quantitatively and appears to be higher closer to the crack. This paper showcases the feasibility and potential of a new experimental approach that enables non-invasive and dynamic mapping of gel fracture mechanics.

Unravelling anisotropic nonlinear shear elasticity in muscles: Towards a non-invasive assessment of stress in living organisms

Acoustoelasticity theory describes propagation of shear waves in uniaxially stressed medium and allows the retrieval of nonlinear elastic coefficients of tissues. In transverse isotropic medium such as muscles the theory leads to 9 different configurations of propagating shear waves (stress axis vs. fibers axis vs. shear wave polarization axis vs. shear wave propagation axis). In this work we propose to use 4 configurations to quantify these nonlinear parameters ex vivo and in vivo. Ex vivo experiments combining ultrasound shear wave elastography and mechanical testing were conducted on iliopsoas pig muscles to quantify three third-order nonlinear coefficients A, H and K that are possibly linked to the architectural structure of muscles. In vivo experiments were performed with human volunteers on biceps brachii during a stretching exercise on an ergometer. A combination of the third order nonlinear elastic parameters was assessed. The knowledge of this nonlinear elastic parameters paves the way to quantify in vivo the local forces produced by muscle during exercise, contraction or movements.

Simultaneous tensile and shear measurement of the human cornea in vivo using S0- and A0-wave optical coherence elastography

Understanding corneal stiffness is valuable for improving refractive surgery, detecting corneal abnormalities, and assessing intraocular pressure. However, accurately measuring the elastic properties, specifically the tensile and shear moduli that govern mechanical deformation, has been challenging. To tackle this issue, we have developed guided-wave optical coherence elastography that can simultaneously excite and analyze symmetric (S0) and anti-symmetric (A0) elastic waves in the cornea at around 10 kHz frequencies, enabling us to extract tensile and shear properties from measured wave dispersion curves. We verified the technique using elastomer phantoms and ex vivo porcine corneas and investigated the dependence on intraocular pressure using acoustoelastic theory that incorporates corneal tension and a nonlinear constitutive tissue model. In a pilot study involving six healthy human subjects aged 31 to 62, we measured shear moduli (Gzx) of 94±20 kPa (mean±standard deviation) and tensile moduli (Exx) of 4.0±1.1 MPa at central corneas. Our preliminary analysis of age-dependence revealed contrasting trends: -8.3±4.5 kPa/decade for shear and 0.30±0.21 MPa/decade for tensile modulus. This OCE technique has the potential to become a highly useful clinical tool for the quantitative biomechanical assessment of the cornea. Statement of SignificanceThis article reports an innovative elastography technique using two guided elastic waves, demonstrating the measurement of both tensile and shear moduli in human cornea in vivo with unprecedented precision. This technique paves the way for comprehensive investigations into corneal mechanics and holds clinical significance in various aspects of corneal health and disease management.

Accurate P-wave reflection and transmission coefficients for non-welded interface incorporating elasto-plastic deformation

P-wave reflection and transmission coefficients for non-welded interface play crucial roles in broad practical engineering productions, involving fracture properties prediction and seismic inversion. However, the existing reflection coefficient equations for non-welded interface in elasto-plastic media are seldom studied, although the elasto-plastic deformation is frequently encountered in the Earth’s subsurface due to artificial and tectonic activities. In this study, we proposed the accurate reflection and transmission coefficients equation for a non-welded interface embedded in an elasto-plastic deformed medium based on the elasto-plastic acoustoelastic and linear-slip theory. In detail, this paper uses elasto-plastic acoustoelastic theory to derive the reflection and transmission coefficients equation. The reflection and transmission coefficients matrix are solved using the linear-slip theory as the boundary condition. Moreover, we use the hardening parameter and plastic deformation to represent the plastic properties of the rock, which is a function of stress and plastic deformation. Through Numerical analysis, the deformation caused by static stress has significantly changed the amplitude and the slope of the reflection and transmission coefficients amplitude. As the stress increases, the rock’s velocity becomes higher, and all reflection and transmission coefficients (i.e., RPP, RPS, TPP, TPS) abruptly change at the critical angle. Furthermore, with the increase in plastic deformation, the critical angle of the incident P-wave and the hardening parameter becomes larger than the unstressed state. The non-welded interface exhibits a low-pass frequency filter for reflected SV-waves and a high-pass frequency filter for reflected P-waves and transmitted P and SV waves. In addition, we can observe that static vertical stress can weaken the anomalous reflections caused by non-welded formations, but the effect is insignificant. On the other hand, the effect of fracture normal compliance to reflection and transmission is detailly investigated. When N&lt;2.5*10-10(MPa-1), The non-welded interface is close to the welded interface, while N&gt;2.5*10-5(MPa-1), the non-welded interface is close to the solid-air interface.

Comparison of ultrasound elastography, magnetic resonance elastography and finite element model to quantify nonlinear shear modulus

Objective. The aim of this study is to validate the estimation of the nonlinear shear modulus () from the acoustoelasticity theory with two experimental methods, ultrasound (US) elastography and magnetic resonance elastography (MRE), and a finite element method. Approach. Experiments were performed on agar (2%)—gelatin (8%) phantom considered as homogeneous, elastic and isotropic. Two specific setups were built to ensure a uniaxial stress step by step on the phantom, one for US and a nonmagnetic version for MRE. The stress was controlled identically in both imaging techniques, with a water tank placed on the top of the phantom and filled with increasing masses of water during the experiment. In US, the supersonic shear wave elastography was implemented on an ultrafast US device, driving a 6 MHz linear array to measure shear wave speed. In MRE, a gradient-echo sequence was used in which the three spatial directions of a 40 Hz continuous wave displacement generated with an external driver were encoded successively. Numerically, a finite element method was developed to simulate the propagation of the shear wave in a uniaxially stressed soft medium. Main results. Similar shear moduli were estimated at zero stress using experimental methods, = 12.3 ± 0.3 kPa and = 11.5 ± 0.7 kPa. Numerical simulations were set with a shear modulus of 12 kPa and the resulting nonlinear shear modulus was found to be −58.1 ± 0.7 kPa. A very good agreement between the finite element model and the experimental models ( = −58.9 ± 9.9 kPa and = −52.8 ± 6.5 kPa) was obtained. Significance. These results show the validity of such nonlinear shear modulus measurement quantification in shear wave elastography. This work paves the way to develop nonlinear elastography technique to get a new biomarker for medical diagnosis.

Effects of confining pressure and pore pressure on multipole borehole acoustic field in fluid-saturated porous media

In-situ stress is a common stress in the exploration and development of oil reservoirs. Therefore, it is of great significance to study the propagation characteristics of borehole acoustic waves in fluid-saturated porous media under stress. Based on the acoustoelastic theory of fluid-saturated porous media, the field equation of fluid-saturated porous media under the conditions of confining pressure and pore pressure and the acoustic field formula of multipole source excitation in open hole are given. The influences of pore pressure and confining pressure on guided waves of multipole borehole acoustic field in fluid-saturated porous media are investigated. The numerical results show that the phase velocity and excitation intensity of guided wave increase significantly under the confining pressure. For a given confining pressure, the phase velocity of the guided wave decreases with pore pressure increasing. The excitation intensity of guided wave increases at low frequency and then decreases at high frequency with pore pressure increasing, except for that of Stoneley wave which decreases in the whole frequency range. These results will help us get an insight into the influences of confining pressure and pore pressure on the acoustic field of multipole source in borehole around fluid-saturated porous media.

Acoustoelastic Mori-Tanaka model for third-order elastic constants of fractured rocks

Insights into the stress-dependent elastic moduli of fractured rocks are important in monitoring geopressure and tectonic stress. This can be addressed by the theory of acoustoelasticity that describes elastic nonlinearity due to the cubic strain-energy function through third-order elastic constants (TOECs). The acoustoelastic TOECs are strictly valid for an isotropic homogeneous medium. The extension to fractured rocks remains largely unaddressed. A group of fractures (ellipses) is embedded into a homogeneous background medium subjected to a uniform confining pressure. We formulate an acoustoelastic Mori-Tanaka (MT) model for the effective TOECs of fractured rocks which can be defined as the weighted average of individual TOECs between the background and fractures in terms of respective volume fractions and stiffnesses. For homogeneously distributed fractures, the stress-induced strains are isotropic in the background and in the fractures. For aligned fractures, the resulting anisotropic strains over the background and fractures make the rock a vertical transverse isotropic medium. The effective elastic moduli and TOECs of fractured rocks depend on background elastic moduli, fracture contents, aspect ratios, and fracture orientations. The prestressed aligned fractures induce nonlinear effects on the effective TOECs that increase with increasing fracture contents and decreasing aspect ratios. We validate the acoustoelastic MT model by experimental data and investigate its limitation by comparing it with finite-element simulations.

Nonlinear constitutive relation in acoustoelastic-orthotropic media and its wide azimuth seismic reflection coefficient characteristics for in-situ stress prediction

With the development of oil and gas exploration targets towards deep water and deep layer, change of subsurface rock physical properties caused by in-situ stress can no longer be ignored. Hence, it is of great significance to study the acoustoelasticity characteristics of rocks under complex subsurface stress environment. Based on acoustoelasticity theory and anisotropic analysis of rock under triaxial stress state, equivalent stiffness matrix of acoustoelastic-orthotropic media is established. According to the equivalent stiffness matrix, exact solution of reflection coefficient for stress-induced orthotropic media is derived. Through model experiments, reflection coefficient varies with incident and azimuth angle is analyzed. Based on the azimuthal characteristics of acoustoelastic medium reflection coefficient and in-situ stress prediction models, two in-situ stress characteristic parameters are proposed to describe in-situ stress distribution. Then, by numerical simulation, the law between azimuthal characteristics of acoustoelastic medium reflection coefficient and in-situ stress characteristic parameters is summarized. And an effective way to predict in-situ stress field that wide azimuth seismic data dependent is established. Finally, through actual wide azimuth seismic data in a field, the practicability of this in-situ stress prediction method rely on stress characteristic parameters is verified. The results illustrate that the prediction method of in-situ stress characteristic parameters can accurately describe the distribution of in-situ stress.

Guided elastic waves in a highly-stretched soft plate

We study the propagation of guided elastic waves in a highly-stretched Ecoflex© plate, a nearly-incompressible elastomer. The plate is subjected to a nearly-uniaxial stress with an elongation reaching 120% and we measure in-plane displacements of the shear horizontal mode SH0 and of the plate mode S0 coexisting in the low frequency limit. An induced anisotropy is first observed and characterized by following the phase velocities in two principal directions. Although these measurements provide an initial stress estimate, we evidence the limits of the acoustoelastic theory to predict those phase velocities in a prestressed elastomer. Taking into account the frequency dependent shear modulus of the elastomer, an experiment-driven fractional rheological model is added to the theory. This provides a proper prediction of phase velocities up to 80% elongation.

Non-equilibrium strain induces hysteresis and anisotropy in the quasi-static and dynamic elastic behavior of sandstones: Theory and experiments

Materials with grain contacts or partially closed cracks exhibit anomalous elastic behavior: hysteresis in quasi-static experiments and slow dynamics in fast dynamic ones. Albeit the behavior in the two cases (which correspond to very different strain ranges) appears different, it should stem from the same physics and, thus, could be modeled by a universal equation of state. We propose a modification of the standard acoustoelastic theory, introducing the concept of conditioning induced non-equilibrium strain, which allows us to predict the evolution of elastic wave velocity in both quasi-static and dynamic ranges, including the velocity anisotropy induced by external uniaxial loading.

Noncontact measurement of bolt axial force in tightening processes using scattered laser ultrasonic waves

This paper presents a new methodology for noncontact measurement of the axial force of bolts in their tightening processes using laser-generated ultrasound waves. This method employs ultrasound waves scattered in a bolt shaft to detect axial force changes, while most conventional ultrasonic methods use ultrasound waves propagating linearly along the bolt axis. The ultrasound waves in this study are generated by laser irradiation on the top surface of a bolt. Subsequently, they propagate deeply into the shaft and return towards the top of the bolt through complicated paths due to the multiple scattering in the shaft. Finally, they are detected at the top surface using another laser and a speckle knife edge detector. With an examination based on the acoustoelastic theory and finite element analysis, we demonstrate that the waveform of the scattered ultrasound shifts linearly with axial force by the cross-correlation method. The proposed technique does not require machining to flatten a bolt's head and the end, while conventional ultrasonic methods need flattening procedures. The proposed technique enables fast, cost-effective axial force measurement in mass production manufacturing processes.

Nondestructive ultrasound evaluation of microstructure-related material parameters of skeletal muscle: an in silico and in vitro study

Direct and nondestructive assessment of material properties of skeletal muscle in vivo shall advance our understanding of intact muscle mechanics and facilitate personalized interventions. However, this is challenged by intricate hierarchical microstructure of the skeletal muscle. We have previously regarded the skeletal muscle as a composite of myofibers and extracellular matrix (ECM), formulated shear wave propagation in the undeformed muscle using the acoustoelastic theory, and preliminarily demonstrated that ultrasound-based shear wave elastography (SWE) could estimate microstructure-related material parameters (MRMPs): myofiber stiffness μf, ECM stiffness μm, and myofiber volume ratio Vf. The proposed method warrants further validation but is hampered by the lack of ground truth values of MRMPs. In this study, we presented analytical and experimental validations of the proposed method using finite-element (FE) simulations and 3D-printed hydrogel phantoms, respectively. Three combinations of different physiologically relevant MRMPs were used in the FE simulations where shear wave propagations in the corresponding composite media were simulated. Two 3D-printed hydrogel phantoms with the MRMPs close to those of a real skeletal muscle (i.e., μf=2.02kPa, μm=52.42kPa, and Vf=0.675,0.832) for ultrasound imaging were fabricated by an alginate-based hydrogel printing protocol that we modified and optimized from the freeform reversible embedding of suspended hydrogels (FRESH) method in literature. Average percent errors of (μf,μm,Vf) estimates were found to be (2.7%,7.3%,2.4%)in silico and (3.0%,8.0%,9.9%)in vitro. This quantitative study corroborated the potential of our proposed theoretical model along with ultrasound SWE for uncovering microstructural characteristics of the skeletal muscle in an entirely nondestructive way.

Effect of axial stresses on longitudinal guided waves in a fluid-filled pipe.

This paper describes the study of the acoustic field of a fluid-filled pipe subjected to axial stress based on the acoustoelastic theory. The pipe with applied axial stresses can be approximated as a transversely isotropic pipe, and hence, its acoustic fields can be expressed using potential functions. The velocity changes of longitudinal wave modes with applied stresses are analyzed for the pipe filled with oil by an analytical method. It was found that the longitudinal mode velocity changes almost uniformly with the applied stresses. The high speed and low frequency plateaus of longitudinal wave modes are sensitive to stress. The relationship between stress and the velocity change of the guided wave is given. The results indicate that non-destructive testing techniques using longitudinal wave modes have strong potential to identify and monitor the stress levels in pipe structures.